Always when I'm allowed on Main Street. Even if they don't use their exhibition rooms for anything interesting, I still like looking around the first storey.
I've had friends specifically show me around their second floor museum exhibits before, but unless they suggest it or I'm in their museum for some reason I usually don't think to look.
Maybe one day when I get my second character's house set up and stop hoarding junk in my museum I'll make my exhibition rooms look ~fancy.
Nah never For the regular part its just the same stuff i have in mine so why would i, and for the exhibit rooms upstairs, i usually completely forget about them because there is just so much to see in the main part of town.
Never, since I've mostly visited people for trades and a couple of times to check out their towns. That could be fun but I wouldn't check it out without asking the other person.
I haven't done anything special with my museum rooms, I just use them to store stuff like gyroids and villager pictures. I've decorated those rooms but not much since they're just for storage. I want to do something nice with them eventually, though.